Put-Call Parity

Question

A stock trades at 102. The 100-strike European call expiring in 3 months trades at 10. Interest rates and dividends are zero. What must the 100-strike put (same expiry) be worth, and why must it?

Accepted Answer

Put-call parity (zero rates/dividends): C - P = S - K implies P = C - S + K = 10 - 102 + 100 = mathbf8. Why it MUST hold, replication, not statistics. Consider buying the call and selling the put (both strike K). At expiry, whatever happens: - S_T > 100: call pays S_T - 100, put expires worthless → you have S_T - 100. - S_T < 100: call worthless, short put costs -(100 - S_T) → again S_T - 100. So call-minus-put is a forward at 100: payoff S_T - 100 in every state. Its price today must equal the forward's value, S - K = 2. If the put traded at 7, you'd sell the call, buy the put, buy the stock (cost 7 - 10 + 102 = 99 net), and deliver at 100 at expiry for a riskless dollar, arbitrage until prices snap back. Interview reflexes this enables: parity converts any call question into a put question instantly; violations of parity are the first thing to scan for in any quoted option chain; and "synthetics" (forward = C - P) are how desks actually think about positions.

Put-Call Parity

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